A race car is driven around a circular tract at a constant speed of 180 miles per hour. If the diameter of the track is 1/2 miles, what is the angular speed of the car? Express your answer in revolutions per hour.
I know the answer is 114.59 revolutions per hour, but I don't know what steps to take to get to that answer. Please show me.
Thanks.
the arclength a of a circle is given by a=rθ where θ is in radians.
differentiate with respect to t
da/dt = .25dθ/dt
180 = .25dθ/dt
dθ/dt = 720 radians per hour
one revolution is 2pi radians, so divide 720 by 2pi to get 114.59 revolutions.
Therefore, the angular speed of the car is 114.59 revolutions per hour.
Well, well, well! Looks like we've got ourselves a speedy race car on a circular track. Let's break it down, shall we?
First, we need to find the angular speed of the car. Remember that angular speed is the rate at which an object rotates around a center point. In this case, that object is our race car zipping around the track.
Now, we're given that the car is moving at a constant speed of 180 miles per hour. But we need to convert that into an angular speed, expressed in revolutions per hour. To do that, we need to use some fancy formulas.
The formula for arc length (a) of a circle is a = rθ, where θ represents the angle in radians. Differentiating both sides of the equation with respect to time (t), we get da/dt = r(dθ/dt). Bear with me, we're almost there!
Since we know the diameter (which is twice the radius) of the track is 1/2 mile, the radius is 1/4 mile. Now, let's plug in the values we have and solve:
180 = 0.25(dθ/dt)
Multiply both sides by 0.25 to isolate dθ/dt:
dθ/dt = 720 radians per hour
Hold your horses, we're almost done! We know that one revolution equals 2π radians. So, to find the angular speed in revolutions per hour, we divide dθ/dt (720 radians per hour) by 2π.
dθ/dt = 720 radians per hour
720 / (2π) ≈ 114.59 revolutions per hour
Voila! The angular speed of the race car is approximately 114.59 revolutions per hour. Now, buckle up and enjoy the ride!
To solve this problem, we can use the formula for arc length. The arc length "a" of a circle is given by the equation a = rθ, where θ is the central angle in radians.
In this case, the diameter of the track is given as 1/2 mile. The radius of the track is half of the diameter, so it is 1/4 mile.
We are given that the car is driving at a constant speed of 180 miles per hour. This speed corresponds to the arc length covered by the car in one hour.
Let's differentiate the arc length equation with respect to time "t" to find the angular speed:
da/dt = r(dθ/dt)
Where da/dt is the derivative of the arc length with respect to time, r is the radius, and dθ/dt is the angular speed.
Substituting the values we know, we get:
180 = (1/4)(dθ/dt)
Now, let's solve for dθ/dt:
Multiply both sides by 4:
180 * 4 = dθ/dt
720 = dθ/dt
The value 720 represents the angular speed of the car in radians per hour.
To convert this to revolutions per hour, we know that one revolution is equal to 2π radians.
So, we divide 720 by 2π:
720 / (2π) ≈ 114.59 revolutions per hour
Therefore, the angular speed of the car is approximately 114.59 revolutions per hour.
To find the angular speed of the race car, we need to use the relationship between the linear speed and the angular speed on a circular track.
The linear speed of the car is given as 180 miles per hour. To convert this to a smaller unit of measurement, we need to convert miles per hour to feet per hour. Since 1 mile equals 5,280 feet, the linear speed of the car becomes 180 * 5280 = 950,400 feet per hour.
Now, let's consider the geometry of the circular track. The diameter of the track is given as 1/2 mile, which corresponds to a radius of 1/4 mile. Since 1 mile equals 5,280 feet, the radius of the track is 1/4 * 5280 = 1,320 feet.
Next, we need to relate the linear speed to the angular speed using the formula:
Linear Speed = Angular Speed * Radius
Plugging in the values, we have:
950,400 feet per hour = Angular Speed * 1,320 feet
To solve for the angular speed, we divide both sides of the equation by 1,320 feet:
Angular Speed = 950,400 feet per hour / 1,320 feet
Angular Speed = 720 revolutions per hour
However, we need to express the angular speed in revolutions per hour, not as a fraction of revolutions per hour.
To convert from radians to revolutions, we know that one revolution is equal to 2π radians. Therefore, we divide the angular speed (720 revolutions per hour) by 2π:
Angular Speed (in revolutions per hour) = 720 revolutions per hour / (2 * 3.14159)
Angular Speed ≈ 114.59 revolutions per hour
So, the angular speed of the race car is approximately 114.59 revolutions per hour.